Elliptic threefolds with high Mordell-Weil rank
The authors present examples of smooth Calabi-Yau \(3\)-folds with the Mordell-Weil (MW) rank \(10\), the highest currently known value, and give detailed results on their properties as models of quantum gravity, such as massless particle spectrum and anomaly cancellation conditions. The MW rank determines the rank of the Abelian gauge algebra in \(F\)-theory compactifications on these \(3\)-folds. The \(3\)-folds \(X_i\) studied in the paper are resolutions of \(\bar{X}:=B\times_{\mathbb{P}^1}B'\), with \(B\), \(B'\) rational elliptic surfaces, first considered by Schoen. The ones with MW rank \(10\) specifically were studied by Namikawa and Rossi for their deformation properties and are known as Namikawa examples. The authors prove that the discriminant of the elliptic fibration \(X_i\to B\) is supported on six cuspidal curves with generic fibers of Kodaira type II that upgrade to Kodaira type IV at the six cusps. It is conjectured that the Namikawa examples have the highest MW rank among the Schoen \(3\)-folds. The paper then develops a detailed geometry-physics dictionary for the Namikawa examples. To compute the massless spectrum and the \(U(1)\) charges, the authors explicitly determine, using extensive calculations, the Poincare pairing between \(H^2\) and \(H_2\), the Shioda map, the height pairings, the Gopakumar-Vafa invariants, and some other invariants. It turns out that charged singlet matter does not have to reside at enhancement loci of type \(I_2\), as previously believed. Geometric counterparts of the gravitational and \(U(1)\) anomaly equations are also derived, and it is verified that the anomaly cancellation conditions are satisfied. Finally, the authors obtain birational invariants of non-\(\mathbb{Q}\)-factorial singularities of the Weierstrass model \(\bar{X}\to B\), and compare the latter to the Weierstrass model \(W_{NDE}\to B\) for \(B=\mathbb{P}^2\) constructed by Elkies. In particular, they explicitly describe a log canonical model \(W_{\mathbb{P}^2}\to\mathbb{P}^2\). This is a step in the direction of establishing whether Elkies models are birationally Calabi-Yau (they are numerically so).
scientific article; zbMATH DE number 7606767